the category of T0 Alexandroff spaces is equivalent to the category of posets
Let 𝒜𝒯 be the category of all T0, Alexandroff spaces and continuous maps between them. Furthermore let 𝒫𝒪𝒮ℰ𝒯 be the category of all posets and order preserving maps.
Theorem. The categories 𝒜𝒯 and 𝒫𝒪𝒮ℰ𝒯 are equivalent.
Proof. Consider two functors:
T:𝒜𝒯→𝒫𝒪𝒮ℰ𝒯; |
S:𝒫𝒪𝒮ℰ𝒯→𝒜𝒯, |
such that T(X,τ)=(X,≤), where ≤ is an induced partial order on an Alexandroff space and T(f)=f for continuous map. Analogously, let S(X,≤)=(X,τ), where τ is an induced Alexandroff topology
on a poset and S(f)=f for order preserving maps. One can easily show that T and S are well defined. Furthermore, it is easy to verify that equalities T∘S=1𝒫𝒪𝒮ℰ𝒯 and S∘T=1𝒜𝒯 hold, which completes
the proof. □
Remark. Of course every finite topological space is Alexandroff, thus we have very nice ,,interpretation” of finite T0 spaces - finite posets (since functors T and S do not change set-theoretic properties of underlying sets such as finitness).
Title | the category of T0 Alexandroff spaces is equivalent to the category of posets |
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Canonical name | TheCategoryOfT0AlexandroffSpacesIsEquivalentToTheCategoryOfPosets |
Date of creation | 2013-03-22 18:46:04 |
Last modified on | 2013-03-22 18:46:04 |
Owner | joking (16130) |
Last modified by | joking (16130) |
Numerical id | 8 |
Author | joking (16130) |
Entry type | Theorem |
Classification | msc 54A05 |